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Farrell–Jones Conjecture
In mathematics, the Farrell–Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms. These maps are given as certain homomorphisms. The motivation is the interest in the target of the assembly maps; this may be, for instance, the algebraic K-theory of a group ring :K_n(RG) or the L-theory of a group ring :L_n(RG), where ''G'' is some group. The sources of the assembly maps are equivariant homology theory evaluated on the classifying space of ''G'' with respect to the family of virtually cyclic subgroups of ''G''. So assuming the Farrell–Jones conjecture is true, it is possible to restrict computations to virtually cyclic subgroups to get information on complicated objects such as K_n(RG) or L_n(RG). The Baum–Connes conjecture formulates a similar statement, for the topological K-theory of reduced group C^*-algebras K^ _n(C^r_*(G)). Formulation One can find for any ring R equivariant homology theories K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lowell E
Lowell may refer to: Places United States * Lowell, Arkansas * Lowell, California * Lowell, Florida * Lowell, Idaho * Lowell, Indiana * Lowell, Bartholomew County, Indiana * Lowell, Maine * Lowell, Massachusetts ** Lowell National Historical Park ** Lowell (MBTA station) ** Lowell Ordnance Plant * Lowell, Michigan * Lowell, North Carolina * Lowell, Washington County, Ohio * Lowell, Seneca County, Ohio * Lowell, Oregon * Lowell, Vermont, a New England town ** Lowell (CDP), Vermont, the main village in the town * Lowell, West Virginia * Lowell (town), Wisconsin ** Lowell, Wisconsin, a village within the town of Lowell * Lowell Hill, California * Lowell Point, Alaska *Lowell Township (other) Other countries * Lowell glacier, near the Alsek River, Canada Elsewhere * Lowell (lunar crater) * Lowell (Martian crater) Institutions in the United States Arizona * Lowell Observatory, astronomical non-profit research institute, Flagstaff California * Lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropy Group
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometry, Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertex (geometry), vertices, the edge (geometry), edges, and the face (geometry), faces of the polyhedron. A group action on a vector space is called a Group representation, representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyahâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surgery Theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold ''M'' of dimension n=p+q+1, could be described as removing an imbedded sphere of dimension ''p'' from ''M''. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. More technically, the idea is to start with a well-understood manifold ''M'' and perform surgery on it to produce a manifold ''M''′ having some desired property, in such a way th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaplansky's Conjecture
The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures. Group rings Let be a field, and a torsion-free group. Kaplansky's ''zero divisor conjecture'' states: * The group ring does not contain nontrivial zero divisors, that is, it is a domain. Two related conjectures are known as, respectively, Kaplansky's ''idempotent conjecture'': * does not contain any non-trivial idempotents, i.e., if , then or . and Kaplansky's ''unit conjecture'' (which was originally made by Graham Higman and popularized by Kaplansky): * does not contain any non-trivial units, i.e., if in , then for some in and in . The zero-divisor conjecture implies the idempotent conjecture and is implied by the unit conjecture. As of 2021, the zero divisor and idempotent conjectures are open. The unit conjecture, however, was disproved for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Benoît Bost
Jean-Benoît Bost (born 27 July 1961, in Neuilly-sur-Seine) is a French mathematician. Early life and education In 1977, Bost graduated from the Lycée Louis-le-Grand and finished first in the Concours général, the national competition for the places at the elite schools. Bost studied from 1979 to 1983 (qualifying in 1981 for the ''agrégation des mathématiques'') at the École Normale Supérieure (ENS), where he was from 1984 to 1988 '' agrégé-préparateur'' (teacher) and worked under the direction of Alain Connes. Career From 1988, Bost was ''chargé de recherches'' and from 1993 ''directeur de recherches'' at CNRS. From 1993 to 2006, he was ''maître de conferences'' at the École polytechnique. He has been a professor at l'Université Paris-Saclay (Paris XI) in Orsay since 1998. Research Bost deals with noncommutative geometry (partly in collaboration with Alain Connes) with applications to quantum field theory, algebraic geometry, and arithmetic geometry. The eponymous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Novikov Conjecture
The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Novikov who originally posed the conjecture in 1965. The Novikov conjecture concerns the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group. According to the Novikov conjecture, the ''higher signatures'', which are certain numerical invariants of smooth manifolds, are homotopy invariants. The conjecture has been proved for finitely generated abelian groups. It is not yet known whether the Novikov conjecture holds true for all groups. There are no known counterexamples to the conjecture. Precise formulation of the conjecture Let G be a discrete group and BG its classifying space, which is an Eilenberg–MacLane space of type K(G,1), and therefore unique up to homotopy equivalence as a CW complex. Let :f\colon M\rightarrow BG be a continuous map from a closed oriented n-dimensional manifold M to BG, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ideal m, and suppose ''a''1, ..., ''a''''n'' is a minimal set of generators of m. Then by Krull's principal ideal theorem ''n'' ≥ dim ''A'', and ''A'' is defined to be regular if ''n'' = dim ''A''. The appellation ''regular'' is justified by the geometric meaning. A point ''x'' on an algebraic variety ''X'' is nonsingular if and only if the local ring \mathcal_ of germs at ''x'' is regular. (See also: regular scheme.) Regular local rings are ''not'' related to von Neumann regular rings. For Noetherian local rings, there is the following chain of inclusions: Characterizations There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if A is a Noetherian local ring with maximal idea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Torsion
In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau(f) which is an element in the Whitehead group \operatorname(\pi_1(Y)). These concepts are named after the mathematician J. H. C. Whitehead. The Whitehead torsion is important in applying surgery theory to non- simply connected manifolds of dimension > 4: for simply-connected manifolds, the Whitehead group vanishes, and thus homotopy equivalences and simple homotopy equivalences are the same. The applications are to differentiable manifolds, PL manifolds and topological manifolds. The proofs were first obtained in the early 1960s by Stephen Smale, for differentiable manifolds. The development of handlebody theory allowed much the same proofs in the differentiable and PL categories. The proofs are much harder in the topological category, requiring the theory of Robion Kirby and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surgery Obstruction
In mathematics, specifically in surgery theory, the surgery obstructions define a map \theta \colon \mathcal (X) \to L_n (\pi_1 (X)) from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when n \geq 5: A degree-one normal map (f,b) \colon M \to X is normally cobordant to a homotopy equivalence if and only if the image \theta (f,b)=0 in L_n (\mathbb pi_1 (X). Sketch of the definition The surgery obstruction of a degree-one normal map has a relatively complicated definition. Consider a degree-one normal map (f,b) \colon M \to X. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve (f,b) so that the map f becomes m-connected (that means the homotopy groups \pi_* (f)=0 for * \leq m) for high m. It is a consequence of Poincaré duality that if we can achieve this for m > \lfloor n/2 \rfloor then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
